Copy animation from rig to rig
If (and only if) both rigs have the same structure, with same bones naming, so you can:


Select the first animated rig.
In the dope sheet window, choose Action Editor context, in drop-down menu at the bottom:

Just to the right, there is an Action browser that allows you to choose the current animation of the selected object. Here you can name your action properly, to find it easily later.

Select the second rig, then in click on the left button of this Action browser to display all existing actions, and select in the list the one you named before.


And that's it! Your second rig should have exactly the same animation as the first one.

Be aware that if you change something in the first rig's animation, it will also change the second's one, since they share the same action. To avoid this, you can click on the "2" button (or any number that shows the number of users you have for the action) to make a single-user copy of the action and have two independent actions: